On the order of accuracy and numerical performance of two classes of finite volume WENO schemes Rui Zhang, Mengping Zhang and Chi-Wang Shu November 29, 29 Abstract In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian meshes. We compare them in terms of accuracy, performance for smooth and shocked solutions, and efficiency in CPU timing. For linear systems both schemes are high order accurate, however for nonlinear systems, analysis and numerical simulation results verify that one of them (Class A) is only second order accurate, while the other (Class B) is high order accurate. The WENO scheme in Class A is easier to implement and costs less than that in Class B. Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes, despite of the difference in their formal order of accuracy. The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows. Key Words: weighted essentially non-oscillatory (WENO) schemes; finite volume schemes; accuracy Dedicated to the memory of Professor David Gottlieb Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 2326, P.R. China. E-mail: rui@ustc.edu.cn. Research supported in part by NSFC grant 879. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 2326, P.R. China. E-mail: mpzhang@ustc.edu.cn. Research supported in part by NSFC grant 679. Division of Applied Mathematics, Brown University, Providence, RI 292, USA. E-mail: shu@dam.brown.edu. Research supported in part by ARO grant W9NF-8--52 and NSF grant DMS-8986.
Introduction and the setup of the schemes In this paper we are interested in numerically solving two dimensional conservation law systems u t + f(u) x + g(u) y = () with suitable initial and boundary conditions, using the finite volume schemes on Cartesian meshes. For this purpose, the computational domain is decomposed to rectangular cells Ω ij = [x i /2, x i+/2 ] [y j /2, y j+/2 ], and for simplicity we assume the mesh sizes x = x i+/2 x i /2 and y = y j+/2 y j /2 are constants. This assumption is not essential: finite volume schemes in this paper can be defined on arbitrary Cartesian meshes, even those with abrupt changes in mesh sizes, without affecting their conservation, accuracy and stability. In a finite volume scheme we seek approximations to the cell averages ũ i,j = x y yj+/2 xi+/2 y j /2 x i /2 u(x, y)dxdy. (2) We use the notation ū to denote the cell averaging operation in the x-direction (integral in the cell [x i /2, x i+/2 ] divided by the cell size x), and ũ to denote the cell averaging operation in the y-direction. The two dimensional cell average ũ can be obtained by successively performing the cell averaging operators in x and in y. If we integrate the conservation law () over the cell Ω ij and then divide by its area, we obtain d ũ i,j dt where ũ i,j is the cell average (2) and + x ( f i+/2,j f i /2,j ) + y (ḡ i,j+/2 ḡ i,j /2 ) = (3) f i+/2,j = y yj+/2 y j /2 f(u(x i+/2, y))dy (4) ḡ i,j+/2 = xi+/2 g(u(x, y j+/2 ))dx (5) x x i /2 are the physical fluxes, which are cell averages of f(u) in y at x = x i+/2 and of g(u) in x at y = y j+/2 respectively. Although (3) looks like a scheme, we should emphasize that it is actually an equality satisfied by the exact solution of the PDE (). Notice that the equality (3) describes the evolution of the cell averages ũ i,j while requiring the information of point values of the solution u in evaluating the physical fluxes in (4) and (5). In order to convert the equality (3) to a scheme (commonly referred to as a finite volume scheme, since it solves for the cell averages ũ i,j rather than point values of the solution) in the following form d ũ i,j dt + x ( ˆf i+/2,j ˆf i /2,j ) + y (ĝ i,j+/2 ĝ i,j /2 ) =, (6) 2
we must use a reconstruction procedure to obtain numerical fluxes ˆf i+/2,j and ĝ i,j+/2 in (6) as accurate approximations to the physical fluxes f i+/2,j and ḡ i,j+/2 in (4) and (5). We consider two classes of finite volume schemes in this paper. Both of them depend crucially on the following one-dimensional reconstruction procedure. One-dimensional reconstruction procedure: Given the cell averages ū i = x xi+/2 x i /2 u(x)dx of a piecewise smooth function u(x), the procedure uses these cell averages in several neighboring cells ū i l,, ū i+k (the collection of these neighboring cells is referred to as the stencil of the reconstruction) to obtain an approximation to the point value u(x i+/2 ) at the cell interface or at another point u(x ) where x is in the cell [x i /2, x i+/2 ]. We require the reconstruction to be high order accurate when u(x) is smooth in the stencil, and to be essentially non-oscillatory near discontinuities. Such reconstruction procedure has been extensively studied in the literature, among high order ones we mention the essentially non-oscillatory (ENO) reconstruction [2] and the weighted ENO (WENO) reconstruction [4, 3]. In the appendix we list the fifth order WENO reconstruction procedure [3] that we use in this paper. More details about WENO reconstruction can be found in, e.g. [7, 8, 5]. For a given physical flux f(u), based on monotonicity in the scalar case and on characteristic information and Riemann solvers in the system case, we can define a numerical flux ˆf(u, u + ) where u and u + approximate the left and right limits of the function u respectively. The numerical flux ˆf(u, u + ) is at least Lipschitz continuous for both arguments, and is consistent with the physical flux in the sense that ˆf(u, u) = f(u). Examples of numerical fluxes for systems of conservation laws in fluid dynamics can be found in, e.g. [, 6]. In this paper we use the Lax-Friedrichs flux and the HLLC flux as representative examples. In the construction of finite volume schemes we often need to use a q-point Gaussian quadrature to approximate line integrals such as those (4) and (5): q q f i+/2,j ω k f(u(x i+/2, yj k )), ḡ i,j+/2 ω k g(u(x k i, y j+/2)) (7) k= where ω k are the Gaussian quadrature weights, and yj k and x k i are the Gaussian quadrature points. In this paper we consider only schemes up to fifth order accuracy, hence q = 3 suffices. We are now ready to define the two classes of finite volume schemes that we will study in this paper. The first class (Class A) of these methods has the following algorithm flowchart: Finite volume scheme, Class A: 3 k=
. Using the two dimensional cell averages ũ i,j, perform a one-dimensional reconstruction procedure in y to obtain a high order accurate approximation to the cell average of u in x at y = y j+/2, denoted by ū i,j+/2. For the purpose of upwinding and stability, typically two different approximations, denoted by ū i,j+/2 and ū + i,j+/2, based on stencils biased to the left and to the right respectively, are obtained. 2. Using the two dimensional cell averages ũ i,j, perform a one-dimensional reconstruction procedure in x to obtain a high order accurate approximation to the cell average of u in y at x = x i+/2, denoted by ũ i+/2,j. Again, two different approximations, denoted by ũ i+/2,j and ũ+ i+/2,j, based on stencils biased to the left and to the right respectively, are obtained. 3. Form the numerical fluxes simply as ˆf i+/2,j = ˆf(ũ i+/2,j, ũ+ i+/2,j ), ĝ i,j+/2 = ĝ(ū i,j+/2, ū+ i,j+/2 ). (8) 4. Obtain the finite volume scheme (6), then discretize it in time by the TVD Runge-Kutta time discretization [9]. We use the third order version in the numerical tests performed in this paper. This finite volume scheme is very simple and efficient, and hence is widely used in applications. However, this scheme is only second order accurate for general nonlinear systems (systems in which the physical fluxes f(u) and g(u) are nonlinear functions of u), regardless of the order of accuracy in the one-dimensional reconstruction procedure used in Steps and 2 above. The problem arises in the choice of the numerical fluxes in Step 3 above. We will give a proof of this fact in the Appendix. Numerical experiments in next section clearly demonstrate this second order accuracy. There is however an important exception. If the system is linear (that is, the physical fluxes f(u) = Au and g(u) = Bu where A and B are constant matrices), such as the linear Maxwell equations and linearized Euler equations, the same high order accuracy as in the one-dimensional reconstruction is achieved, see the Appendix for a proof and next section for numerical verification. The second class (Class B) of the finite volume methods that we will study in this paper has the following algorithm flowchart: Finite volume scheme, Class B:. Using the two dimensional cell averages ũ i,j, perform a one-dimensional reconstruction procedure in y to obtain a high order accurate approximation to the cell average of u in x at y = y j+/2, denoted by ū i,j+/2. For the purpose of upwinding and stability, typically two different approximations, denoted by ū i,j+/2 and ū+ i,j+/2, based on stencils biased to the left and to the right respectively, are obtained. This step is the same as the first step for the finite volume scheme in Class A. 4
2. Using the one dimensional cell averages ū ± i,j+/2, perform a one-dimensional reconstruction procedure in x to obtain a high order accurate approximation to the point values of u at the Gaussian quadrature points x k i, denoted by u ± i k,j+/2. 3. Using the two dimensional cell averages ũ i,j, perform a one-dimensional reconstruction procedure in x to obtain a high order accurate approximation to the cell average of u in y at x = x i+/2, denoted by ũ i+/2,j. Again, two different approximations, denoted by ũ i+/2,j and ũ+ i+/2,j, based on stencils biased to the left and to the right respectively, are obtained. This step is the same as the second step for the finite volume scheme in Class A. 4. Using the one dimensional cell averages ũ ± i+/2,j, perform a one-dimensional reconstruction procedure in y to obtain a high order accurate approximation to the point values of u at the Gaussian quadrature points yj k, denoted by u ± i+/2,j k. 5. Form the numerical fluxes using the Gaussian quadrature (7): ˆf i+/2,j = q k= ω k ˆf(u i+/2,j k, u + i+/2,j k ), ĝ i,j+/2 = q ω k ĝ(u i k,j+/2, u+ i k,j+/2 ). k= (9) 6. Obtain the finite volume scheme (6), then discretize it in time by the TVD Runge-Kutta time discretization [9]. Comparing with the finite volume scheme in Class A, we can observe that the finite volume scheme in Class B is more complicated and more costly, because of the additional reconstructions in Steps 2 and 4 and the Gaussian quadrature sums in Step 5. However, this scheme is genuinely high order accurate for nonlinear systems. We will give a proof of this fact in the Appendix. Numerical experiments in next section clearly demonstrate the high order accuracy. 2 Numerical experiments We use the two dimensional Euler equations for compressible gas dynamics, namely () with ρ ρv ρw u = ρv ρw, f(u) = ρv 2 + p ρvw, g(u) = ρvw ρw 2 + p, E v(e + p) w(e + p) where ρ is the density, (v, w) is the velocity, E is the total energy, E = + γ 2 ρ(v2 + w 2 ) where p is the pressure, with γ =.4 in our computation. The two classes p 5
of finite volume WENO schemes considered in the previous section are compared for accuracy (both linear and nonlinear cases), resolution for shocked solutions, and CPU costs. Time discretization is by the third order TVD Runge-Kutta method [9], with a suitably reduced time step for the accuracy tests so that spatial errors dominate. Example. The initial condition is set to be ρ(x, y, ) = +.5 sin(π(x + y)) v(x, y, ) = w(x, y, ) = p(x, y, ) =. with periodic boundary conditions. The exact solution is ρ(x, y, ) = +.5 sin(π(x + y 2t)) v(x, y, ) = w(x, y, ) = p(x, y, ) =. () () It is easy to check that, in this special case, the Euler equations actually become a linear system. The computational domain is taken as [, ] [, ] with periodic boundary conditions. We use uniform meshes and denote x = y = h, and give the results with both the Lax-Friedrichs and the HLLC fluxes at time t = 2. in Tables and 2. Clearly, for this linear system case both classes of the finite volume schemes are fifth order accurate. Table : Example. L errors and orders of accuracy for density. Class A finite volume scheme. Lax-Friedrichs Flux HLLC Flux h CFL Error Order Error Order 2/2.9 7.3E-4 7.25E-4 2/4.8 2.36E-5 4.95 2.34E-5 4.95 2/8.6 7.6E-7 4.96 7.53E-7 4.96 2/6.5 2.6E-8 4.87 2.58E-8 4.87 2/32.3 7.95E- 5.3 7.89E- 5.3 This example is often used in the literature to test accuracy of schemes. As we can see here, this is not a stringent test case. The Class A finite volume scheme, which is only second order accurate for general nonlinear problems, easily passes this test as a fifth order scheme. The CPU time used for both classes of the finite volume WENO schemes is listed in Table 3. Here and below the CPU time is measured on a.5ghz Itanium 2 Madison 6
Table 2: Example. L errors and orders of accuracy for density. Class B finite volume scheme. Lax-Friedrichs Flux HLLC Flux h CFL Error Order Error Order 2/2.9.87E-3 7.25E-4 2/4.8 6.2E-5 4.93 2.34E-5 4.95 2/8.6.94E-6 4.98 7.53E-7 4.96 2/6.5 6.28E-8 4.95 2.58E-8 4.87 2/32.3.95E-9 5. 7.88E- 5.3 64 bit CPU. We can see that the Class B finite volume WENO scheme uses about 3 to 4 times more CPU time than the Class A scheme on the same mesh. This is consistent with operation counts of these two classes of schemes. Table 3: Example. CPU time (in seconds) for both classes of finite volume schemes. Lax-Friedrichs Flux HLLC Flux h Class A Class B Ratio Class A Class B Ratio 2/2.77 4.3 6.9 4.3 2/4 3.5 5.25 4.34 3.96 6.33 4.2 2/8 36.97 58.5 4.29 4.57 7.32 4. 2/6 35.63 54.3 4.28 396.45 65.4 4.7 2/32 64.95 234.8 3.38 664.45 2856. 3.29 Example 2 (2D vortex evolution). This example [7] is a nonlinear example, for which we can observe different orders of accuracy for the two classes of finite volume WENO schemes. The setup of the problem is as following. The mean flow is ρ =, p =, (v, w) = (, ), and the computational domain is [, ] [, ]. We add, to the mean flow, an isentropic vortex, which corresponds to perturbations in (v, w) and the temperature T = p ρ and no perturbation in the entropy S = p ρ γ : (δv, δw) = ε (γ )ε2 2π e.5( r2) ( ȳ, x), δt = e r2, δs =, 8γπ 2 where ( x, ȳ) = (x 5, y 5), r 2 = x 2 + ȳ 2, and the vortex strength ε = 5. Periodic boundary condition is used, which has little effect on the solution since the vortex is close to zero at the initial boundary. It is clear that the exact solution is just the passive convection of the vortex with the mean velocity. The errors at time t =.2 calculated by the two classes of finite volume WENO schemes are listed in Tables 4 and 5, respectively. We can observe clearly that the accuracy for the Class A finite 7
Table 4: Example 2. L errors and orders of accuracy for density. Class A finite volume scheme. Lax-Friedrichs Flux HLLC Flux h CFL Error Order Error Order /4.8 6.38E-5 5.45E-5 /8.6 5.24E-6 3.6 4.5E-6 3.6 /6.5 5.8E-7 3.7 5.67E-7 2.99 /32.3.38E-7 2.7.38E-7 2.4 /64. 3.46E-8 2. 3.46E-8 2. Table 5: Example 2. L errors and orders of accuracy for density. Class B finite volume scheme. Lax-Friedrichs Flux HLLC Flux h CFL Error Order Error Order /4.8 7.99E-5 5.39E-5 /8.6 6.54E-6 3.6 3.98E-6 3.76 /6.5 2.2E-7 4.9.2E-7 5.6 /32.3 4.83E-9 5.5 2.69E-9 5.48 /64..3E- 5.2 7.44E- 5.8 volume scheme is only second order, while the accuracy for the Class B scheme is fifth order. Example 3. We add suitable source terms to the right-hand side of the Euler equations such that the following smooth functions are exact solutions of the resulting nonlinear PDE system ρ(x, y, t) = +.5 sin(π(x + y 3t)) v(x, y, t) = cos(π(x + 2y + 2t)) w(x, y, t) =.5 sin(π(2x + y 3t)) p(x, y, t) = +.5 sin(π(x y + 4t)). The computational domain is [, ] [, ] and periodic boundary condition is applied. The error at time t =. calculated by the two classes of finite volume WENO schemes are listed in Tables 6 and 7, respectively. We observe once more that the accuracy for the Class A finite volume scheme is only second order, while the accuracy for the Class B scheme is fifth order. In Figure, we plot the relationship between the L error and CPU time for both classes of finite volume schemes in this example. Even though the Class B finite volume WENO scheme uses about 3 to 4 times more CPU time than the Class A scheme on the same mesh, to reach a given error tolerance below 2. 3, the Class B scheme uses less CPU time. This is consistent with the general conclusions 8
Table 6: Example 3. L errors and orders of accuracy for density. Class A finite volume scheme. Lax-Friedrichs Flux HLLC Flux h CFL Error Order Error Order 2/2.8 3.27E-2 3.6E-2 2/4.6 6.9E-3 2.4 5.83E-3 2.39 2/8.5.2E-3 2.37.9E-3 2.29 2/6.3 3.6E-4.98 3.7E-4.96 2/32. 7.77E-5.98 7.78E-5.98 Table 7: Example 3. L errors and orders of accuracy for density. Class B finite volume scheme. Lax-Friedrichs Flux HLLC Flux h CFL Error Order Error Order 2/2.8 4.42E-2 2.66E-2 2/4.6 5.6E-3 2.98 3.75E-3 2.83 2/8.5 3.82E-4 3.88.8E-4 4.38 2/6.3.43E-5 4.74 6.29E-6 4.85 2/32. 4.92E-7 4.86 2.26E-7 4.8 about high order schemes: they are more economical when high precision simulation is required. Example 4 (Oblique Lax shock tube). The purpose of this test is to assess the capability of the two classes of WENO schemes in resolving waves that are oblique to the computational mesh. The two dimensional Lax problem is solved where the initial jump makes an angle θ against x-axis ( < θ π/2). If θ = π/2, we have the one-dimensional Lax problem. If < θ < π/2, all the waves produced will be oblique to the rectangular computational mesh. We take our computational domain to be [ 7, 7] [ 7, 7], and θ = π/4. The initial jump starts at (x, y) = ( 7, 7). That is, initially we have { (45,.698 cos( π (ρ, v, w, p) = ),.698 sin( π ), 3.528), if x < y 4 4 (.5,,,.57), otherwise (2) The solution is computed up to t =.3 for the mesh size x = y = /3. We show in Figure 2 the densities at y =. The solid lines are the exact solution. We can observe that both classes of the finite volume WENO scheme give visually similar results on the same meshes, despite of their different formal order of accuracy. Example 5 (A Mach 3 wind tunnel with a step). This model problem has been carefully examined in Woodward and Colella [] and later by many others. The 9
. Class A WENO Class B WENO. Class A WENO Class B WENO.... Error. Error. e-5 e-5 e-6 e-6 e-7. CPU time(seconds) e-7. CPU time(seconds) Figure : Example 3. L error versus CPU time for both classes of WENO schemes. Left: Lax-Friedrichs flux; Right: HLLC flux. setup of the problem is the following. The wind tunnel is length unit wide and 3 length units long. The step is.2 length unit high and is located.6 length units from the left-hand end of the tunnel. The problem is initialized by a right going Mach 3 flow. Reflective boundary conditions are applied along the walls of the tunnel and in-flow and out-flow boundary conditions are applied at the entrance (left-hand end) and the exit (right-hand end). For the treatment of the singularity at the corner of the step, we adopt the same technique used in [], which is based on the assumption of a nearly steady flow in the region near the corner. The numerical results are shown in Figure 3, for the mesh sizes x = y = 6 and. We can observe that both 32 classes of the finite volume WENO scheme give visually similar results on the same meshes, despite of their different formal order of accuracy. Example 6 (Double Mach reflection of a strong shock). The computational domain for this problem is chosen to be [, 4] [, ]. The reflecting wall lies at the bottom of the computational domain starting from x = /6. Initially a right-moving Mach shock is positioned at x = /6, y =, and makes a 6 angle with the x-axis. For the bottom boundary, the exact postshock condition is imposed for the part from x = to x = /6 and a reflective boundary condition is used for the rest. At the top boundary of our computational domain, the flow values are set to describe the exact motion of the Mach shock. See [] for a detailed description of the problem. We give the results of both classes of the finite volume WENO schemes in Figure 4 with x = y = /48. The results with x = y = /96 are given in Figure 5. Again, we can observe that both classes of the finite volume WENO scheme give comparable resolutions on the same meshes, despite of their different formal order of accuracy.
.3.4.4.3.3.2.2...9.9.8.8.7.7.6.6.5.5 2 4 6 8 2 4.3 2 4 6 8 2 4 Figure 2: Oblique Lax shock tube. Density at y =. x = y = /3. Left: Class A WENO; Right: Class B WENO. 3 Concluding remarks Two classes of finite volume WENO schemes are discussed and tested in this paper. We observe numerically and prove theoretically that they are both high order accurate for linear systems, however the Class A schemes are only second order accurate for nonlinear systems, while the Class B schemes are still high order accurate. The operation count and CPU timing for the Class B schemes are about 3 to 4 times of those for the Class A schemes on the same mesh. However, for high precision simulation of smooth flows, Class B schemes could take less CPU time to reach the same error threshold than Class A schemes. For problems with shocks, at least for the test problems that we have experimented in this paper, the two classes of finite volume schemes give comparable resolution on the same meshes, despite of their difference in formal order of accuracy. Results in this paper might shed some light on the popularity of Class A schemes in applications. A Appendix In this appendix we describe briefly the fifth order WENO reconstruction procedure, and prove the orders of accuracy for the two classes of WENO schemes studied in this paper.
.5.5 2 2.5 3.9.8.7.6.5.3.2. (a) x = y =, Class A WENO 6.9.8.7.6.5.3.2..5.5 2 2.5 3 (b) x = y =, Class B WENO 6.9.8.7.6.5.3.2..5.5 2 2.5 3 (c) x = y =, Class A WENO 32.9.8.7.6.5.3.2..5.5 2 2.5 3 (d) x = y =, Class B WENO 32 Figure 3: Flow past a forward facing step. 3 contours from. to 6.6, Lax-Friedrichs flux. 2
.8.6.2.5.5 2 2.5 3 (a) Class A WENO.8.6.2.5.5 2 2.5 3 (b) Class B WENO.5.5.3.3.2.2.. 2 2.2 2.4 2.6 2.8 3 2 2.2 2.4 2.6 2.8 3 (c) Blow-up regions near the Mach stem for details. Left: Class A WENO; Right: Class B WENO Figure 4: Result for the double Mach reflection. x = y =, Class A WENO 48 (a) and Class B WENO (b), HLLC flux. 3
.8.6.2.5.5 2 2.5 3 2 2.5 3 (a) Class A WENO.8.6.2.5.5 (b) Class B WENO.5.5.3.3.2.2.. 2 2.2 2.4 2.6 2.8 3 2 2.2 2.4 2.6 2.8 3 (c) Blow-up regions near the Mach stem for details. Left: Class A WENO; Right: Class B WENO Figure 5: Result for the double Mach reflection. x = y = (a) and Class B WENO (b), HLLC flux. 4, Class A WENO 96
A. Fifth order WENO reconstruction We use the fifth-order WENO reconstruction procedure, described in [3]. Lower or higher order reconstructions are also possible, see for example those in [4, ]. For a piecewise smooth function v(x), we denote v i as its cell average. The fifthorder accurate reconstruction with a left-biased stencil is defined as v j+/2 = ω v + ω v + ω 2 v 2 (3) where v k are the reconstructed values obtained from cell averages in the k-th stencil S k = (j k, j k +, j k + 2): v = 6 ( v i+2 + 5 v i+ + 2 v i ), v = 6 (2 v i+ + 5 v i v i ), (4) v 2 = 6 ( v i 7 v i + 2 v i 2 ) and ω k, k =, 2, 3 are the nonlinear WENO weights given by ω k = α k 3 l= α, α = l.3 (ε + IS ) 2, α =.6 (ε + IS ) 2, α 2 =. (ε + IS 2 ) 2. (5) Here ε is a small parameter which we take as ε = 6 in our numerical tests. The smoothness indicators IS k are given by [3] IS = 3 2 ( v i 2 v i+ + v i+2 ) 2 + 4 (3 v i 4 v i+ + v i+2 ) 2, IS = 3 2 ( v i 2 v i + v i+ ) 2 + 4 ( v i + v i+ ) 2, IS 2 = 3 2 ( v i 2 2 v i + v i ) 2 + 4 (3 v i 2 4 v i + 3 v i ) 2. (6) The right-biased reconstruction v + i+/2 is obtained by symmetry with respect to the location i+/2. Reconstructions to values v(x ) for x inside the interval [x i /2, x i+/2 ] can be obtained along the same lines, in fact only the formulas for the reconstructed values obtained from the small stencils in (4) and the linear weights (.3,.6 and.) in (5) would change. We refer to, e.g. [5] for more details. For the systems of conservation laws, a local characteristic decomposition is used in the reconstruction. We refer to [3, 7] for more details. A.2 Accuracy for the two classes of finite volume WENO schemes Theorem A.. The Class B finite volume WENO scheme is accurate to the order of accuracy of the reconstruction and quadrature rules. 5
Proof: We take as an example the case of a fifth order reconstruction and three point quadrature rules used in this paper. The proof for the general case is the same. The values u ± i+/2,j k at the Gaussian points in (9) are reconstructed by the fifth-order WENO procedure, which yields u ± i+/2,j k = u(x i+/2, y k j ) + O( x5 + y 5 ) (7) where u(x, y) is the exact solution of the PDE (we have suppressed the t variable for simplicity, as we are considering truncation errors in space only). Since the numerical flux ˆf is Lipschitz continuous, we have ˆf(u i+/2,j k, u + i+/2,j k ) f(u(x i+/2, y k j )) = ˆf(u i+/2,j k, u + i+/2,j k ) ˆf(u(x i+/2, y k j ), u(x i+/2, y k j )) ˆf(u i+/2,j k, u + i+/2,j k ) ˆf(u i+/2,j k, u(x i+/2, y k j )) + ˆf(u i+/2,j k, u(x i+/2, y k j )) ˆf(u(x i+/2, y k j ), u(x i+/2, y k j )) L( u + i+/2,j k u(x i+/2, y k j ) + u i+/2,j k u(x i+/2, y k j ) ) = O( x 5 + y 5 ). (8) The error for the 3-point Gaussian integration in (7) is f i+/2,j = 3 ω k f(u(x i+/2, yj k )) + O( y6 ). (9) k= So we can get the error of the flux ˆf i+/2,j in (9) as f i+/2,j = ˆf i+/2,j + O( x 5 + y 5 ) (2) Assuming the leading term in the error O( x 5 + y 5 ) in (2) is smooth, we have x ( f i+/2,j f i /2,j ) x ( ˆf i+/2,j ˆf i /2,j ) = O( x 5 + y 5 ). (2) Similarly, we have y (ḡ i,j+/2 ḡ i,j /2 ) y (ĝ i,j+/2 ĝ i,j /2 ) = O( x 5 + y 5 ). (22) We can see from (2) and (22) that the local truncation error for the Class B WENO scheme is fifth order. Theorem A.2. The Class A finite volume WENO scheme is second order accurate for general nonlinear conservation laws, but it is accurate to the order of the reconstruction and quadrature rules for linear conservation laws with constant coefficients. 6
Proof: The values ũ ± i+/2,j in (8) are reconstructed by the fifth order WENO procedure, hence we have ũ ± i+/2,j = ũ i+/2,j + O( x 5 ). (23) By the Lipschitz continuity of the numerical flux ˆf, we have ˆf i+/2,j f(ũ i+/2,j ) = ˆf(ũ i+/2,j, ũ+ i+/2,j ) ˆf(ũ i+/2,j, ũ i+/2,j ) ˆf(ũ i+/2,j, ũ+ i+/2,j ) ˆf(ũ i+/2,j, ũ i+/2,j) + ˆf(ũ i+/2,j, ũ i+/2,j) ˆf(ũ i+/2,j, ũ i+/2,j ) L( ũ i+/2,j ũ i+/2,j + ũ + i+/2,j ũ i+/2,j ) = O( x 5 ). (24) That is ˆf i+/2,j = f(ũ i+/2,j ) + O( x 5 ). (25) Therefore we have ˆf i+/2,j ˆf i /2,j x = f(ũ i+/2,j) f(ũ i /2,j ) x + O( x 5 ). (26) Notice that the cell average of a function agrees with its value at the center of the cell to second order: ũ i+/2,j = u(x i+/2, y j ) + O( y 2 ), (27) and the combination of which gives f i+/2,j = f(u(x i+/2, y j )) + O( y 2 ), (28) f(ũ i+/2,j ) = f(u(x i+/2, y j ) + O( y 2 )) = f(u(x i+/2, y j )) + O( y 2 ). (29) We now have f i+/2,j = f(ũ i+/2,j ) + O( y 2 ), (3) which is the crucial source of second order error regardless of the order of reconstruction. We have, therefore, x ( f i+/2,j f i /2,j ) x ( ˆf i+/2,j ˆf i /2,j ) = x ( f i+/2,j f i /2,j ) x (f(ũ i+/2,j) f(ũ i /2,j )) + O( x 5 ) = x ( f i+/2,j f(ũ i+/2,j )) x ( f i /2,j f(ũ i /2,j )) + O( x 5 ) = O( x 5 ) + O( y 2 ) (3) where ˆf i+/2,j is defined with (8). 7
Similarly, y (ḡ i,j+/2 ḡ i,j /2 ) y (ĝ i,j+/2 ĝ i,j /2 ) = O( x 2 ) + O( y 5 ). (32) From (3) and (32) we can see that the scheme is only second order accurate. Now, if f(u) = Au is linear, then ( yj+/2 ) yj+/2 f(u(x i+/2, y))dy = f u(x i+/2, y)dy, (33) y j /2 that is, instead of (3) we now have y j /2 f i+/2,j = f(ũ i+/2,j ), (34) thereby avoiding the crucial second order error. From (3) we can then get x ( f i+/2,j f i /2,j ) x ( ˆf i+/2,j ˆf i /2,j ) = O( x 5 ) (35) which leads to fifth order in the local truncation error for this linear case. References [] D.S. Balsara and C.-W. Shu, Monotonicity preserving weighted essentially nonoscillatory schemes with increasingly high order of accuracy, Journal of Computational Physics, 6 (2), 45-452. [2] A. Harten, B. Engquist, S. Osher and S. Chakravarthy, Uniformly high order essentially non-oscillatory schemes, III, Journal of Computational Physics, 7 (987), 23-33. [3] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 26 (996), pp.22-228. [4] X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 5 (994), 2-22. [5] Y.-Y. Liu, C.-W. Shu and M. Zhang, On the positivity of linear weights in WENO approximations, Acta Mathematicae Applicatae Sinica, 25 (29), 53-538. [6] J. Qiu, B.C. Khoo and C.-W. Shu, A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes, Journal of Computational Physics, 22 (26), 54-565. 8
[7] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, volume 697, Springer, Berlin, 998, 325-432. [8] C.-W. Shu, High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Review, 5 (29), 82-26. [9] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77 (988), 439-47. [] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, a Practical Introduction, Springer, Berlin, 997. [] P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of Computational Physics, 54 (984), 5-73. 9